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Student[Statistics][ShapiroWilkWTest] Overview

overview of the Shapiro Wilks W-Test

 

Description

Examples

Description

• 

Shapiro and Wilk's W-test is a test for normality. The Shapiro Wilk test tests the null hypothesis that a sample follows a normal distribution.

• 

The formula of the test statistic is:

W=i=1naiXi2n1VarianceX

  

where X is the studied sample, Xi is the ith smallest data in X, Xi is the ith data in X, ai are the coefficients to estimate straightness of the quantile-quantile plot.

  

The definitions of these coefficients are beyond the scope of this guide.

• 

The null hypothesis that the sample follows a normal distribution is rejected if W is too small.

Examples

Pete wants to use a one sample t-test to test the mean of the average lifetime of light bulbs of a particular type, but he does not know if the observations are normally distributed. To test this, he applies Shapiro and Wilk's W-test to the sample of data.

His observed data:

 

bulb1

bulb2

bulb3

bulb4

bulb5

bulb6

bulb7

bulb8

bulb9

bulb10

lifetime(hrs)

355.0

359.5

379.3

366.5

325.1

334.4

308.4

355.6

381.2

316.9

 

bulb11

bulb12

bulb13

bulb14

bulb15

bulb16

bulb17

bulb18

bulb19

bulb20

lifetime(hrs)

379.0

338.7

380.3

366.4

368.1

333.3

390.7

337.4

373.3

370.0

Determine the null hypothesis:

  

Null hypothesis: The data is normally distributed

Collect the data:

X355.0,359.5,379.3,366.5,325.1,334.4,308.4,355.6,381.2,316.9,379.0,338.7,380.3,366.4,368.1,333.3,390.7,337.4,373.3,370.0:

Run the Shapiro Wilk w-Test:

Student:-Statistics:-ShapiroWilkWTestX:

Shapiro and Wilk's W-Test for Normality
---------------------------------------
Null Hypothesis:
Sample drawn from a population that follows a normal distribution
Alt. Hypothesis:
Sample drawn from population that does not follow a normal distribution
 
Sample Size:             20
Computed Statistic:      .935508635130523
Computed p-value:        .207505438819378
 
Result: [Accepted]
This statistical test does not provide enough evidence to conclude that the null hypothesis is false.

The Shapiro and Wilk's W-test returns a p-value = 0.207505. From this p-value, Pete concludes that the data can indeed be assumed to be normal and proceed with one sample t-test.

See Also

Student[Statistics][ShapiroWilkWTest]